Should I bother cancelling if the traveller is planning not to show up for a flight?

Non-refundable doesn't mean what you seem to assume - it does not mean that the money is lost. It means that you don't get it back.

That might seem an irrelevant difference, but often it is not - you can use the value of the ticket for further bookings with the airline (for the same flier, not just anybody).
If you cancel the ticket, you typically have 365 days (from day of payment, not day of flight) to use the value for another booking.

The no-show fee is a fee that gets taken out of your remaining value - if the flier simply doesn't show up (instead of cancelling), he will not get the full ticket value parked, but it will be reduced by the no-show fee.
So you should definitely cancel, as it will save the no-show fee. Only if the person is not going to fly within a full year is the ticket value completely lost.

Yes, cancel the ticket, especially since it's someone else's money that bought it. Cancelling requires no more than a few minutes of your time, perhaps less time than you needed to post this question, and even if you get no money out of it, it is courteous to the airline and to anyone else who might want to reserve a seat on that flight. On top of that, there's a chance, however small, that the airline will refund a greater portion of the fare for a cancellation than they would for a no-show.

As pointed out by user Azor Ahai in a helpful comment, you may be entitled to a refund of certain taxes and fees even if the ticket is not refundable.

Cancel the ticket

There is really no downside to cancelling with the airline but there may be some significant potential upside.

Read up on the exact terms and conditions of the ticket and the rules of the specific airline. "Not refundable" doesn't always mean you get nothing back. United, for example, allows you to bank the money for future travel although you need to pay a "change fee" with new money, to use the stored amount. In most cases that's significantly better than just losing all of it.

In most cases you are also entitled to refund of certain taxes and fees that you won't incur.

No-show penalty is typically taken out of the original ticket price/refund. No-show penalty + change fee will often wipe out the price of the entire ticket.

Personally I would notify the airline about the change of plans because that will allow the airline to adjust their passenger manifest prior to the flight and not waste time/effort trying to locate the student when the flight is at the gate. It also allows the airline to fill the students seat with another passenger.

I am not familiar with the "No Show" fees. I did a quick google and came up with No show fee vs Cancelling? on TripAdvisor, and the suggestion there is not so much that you are charged the fee, but rather that amount is subtracted from the value of the ticket if you are asking for a refund or change of date. That aspect could be clarified by someone else.

But that leads to another reason for calling the airline. You could possibly change the flight (with fees) for up to one year later and benefit at least from some of the sunk cost of this ticket. However the ticket would still be tied to the student.

There was an interesting discussion about that latter point on the Workplace SE Being asked to return flight reimbursement after conference was cancelled and we recieved flight credit

Several airlines have announced that they will waive change fees for tickets bought after the coronavirus outbreak. If the student calls to cancel, they may get the full value of the ticket as a credit towards future flights.

In some cases the ticket can be changed instead of cancelled.

For example, I had a flight due with Singapore Airlines on work-related business. The work meeting was cancelled. I phoned up Singapore Airlines and told them this, and I was informed that because there isn't an official ban for that destination, and the ticket is non-refundable, I cannot get the money back.

However, they did offer to cancel the current flights and keep the booking so when I do eventually fly, they can give me new dates. All I have to pay is a small administration fee (several tens of dollars), and the price difference at the time, if such exists.

Contact the airline and discuss the options. Don't just disappear.

Answering your second question, without the money aspect :

Having a passenger not show up puts some stress on airport staff as they'll try to find that person (announcements, double checking seats, passengers that checked in, etc.). This can also reflect on them negatively depending on their manager.

Cancelling the ticket means that you'll spare them the stress. You'll also spare the other passengers potential wait as the pilots will sometimes wait a few more minutes in case the no-showers are simply late.

Ignoring the money aspect, it'd be a nice gesture to cancel as you'll make the flight more comfortable for everyone else involved.

So yes, I would say you should bother cancelling with the airline.

(source: My girlfriend works as a boarding/check-in agent. )

I see it as an economics problem. Other last-minute travellers gain the opportunity to board a flight if I cancel. The airline gains some money (or, at least, it has a certain probability of selling the ticket and gaining money) if I cancel a flight I am not boarding.

However, none of that money gets passed back to me. So why should I opt to help the airline just out of goodwill? If the airline decides to change their terms and conditions and starts offering me money for cancelling, then I'm listening. They are the ones that started monetizing everything heavily (from flight prices to custom seating to business-class upgrades), so I don't feel bad at all. I'm just playing their game.

I would not feel bad at all in calling the company and asking them what they can offer me if I cancel my flight.

I decided to cancel the airline tickets, and I informed our travel agent about this.

Initially, she told me that:

Please be note that:

1. The travel voucher will have to be used under the same passenger
2. Travel voucher will be valid for 1 year from the date of issue
3. New booking will have to reprice for the air fare, subject to fare and tax difference, airline terms and conditions applied.

Later, I got an email saying:

Just receive the updates from our airline contact, due to current situation and uncertainty for future travel, there will be a high chance to allow cancel and refund ticket that involves Spain.

Kindly acknowledge a ticket refund handling fee of (roughly 20 USD) will be charge per ticket in this case.

So in this case, it seems that due to the deteriorating COVID-19 situation in Spain, we will probably be allowed to cancel our tickets and receive a refund, minus the ticket refund handling fee charged by the travel agent.